Question: The lifespans of seals in a particular zoo are normally distributed. The average seal lives $15.8$ years; the standard deviation is $3.8$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living longer than $12$ years.
$15.8$ $12$ $19.6$ $8.2$ $23.4$ $4.4$ $27.2$ $68\%$ $16\%$ $16\%$ We know the lifespans are normally distributed with an average lifespan of $15.8$ years. We know the standard deviation is $3.8$ years, so one standard deviation below the mean is $12$ years and one standard deviation above the mean is $19.6$ years. Two standard deviations below the mean is $8.2$ years and two standard deviations above the mean is $23.4$ years. Three standard deviations below the mean is $4.4$ years and three standard deviations above the mean is $27.2$ years. We are interested in the probability of a seal living longer than $12$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $68\%$ of the seals will have lifespans within 1 standard deviation of the average lifespan. The remaining $32\%$ of the seals will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({16\%})$ will live less than $12$ years and the other half $({16\%})$ will live longer than $19.6$ years. The probability of a particular seal living longer than $12$ years is ${68\%} + {16\%}$, or $84\%$.